Global Bases for Nonplanar Loop Integrands, Generalized Unitarity, and the Double Copy to All Loop Orders

TL;DR

This paper presents a new graph-based method to construct a global basis for nonplanar loop integrands in scattering amplitudes, simplifying unitarity cuts and enabling the application of the double copy at all loop orders.

Contribution

It introduces a constructive, non-redundant integrand basis that streamlines generalized unitarity and facilitates the double copy for gravitational amplitudes at any loop order.

Findings

Provides a global basis for nonplanar integrands

Enables direct reading of coefficients from cuts

Facilitates double copy construction at all loops

Abstract

We introduce a constructive method for defining a global loop-integrand basis for scattering amplitudes, encompassing both planar and nonplanar contributions. Our approach utilizes a graph-based framework to establish a well-defined, non-redundant basis of integrands. This basis, constructed from a chosen set of non-redundant graphs together with a selection of irreducible scalar products, provides clear insights into various physical properties of scattering amplitudes and proves useful in multiple contexts, such as on-shell Ward identities and manifesting gauge-choice independence. A key advantage of our integrand basis is its ability to streamline the generalized unitarity method. Specifically, we can directly read off the coefficients of basis elements without resorting to ansatze or solving linear equations. This novel approach allows us to lift generalized unitarity cuts -- expressed as products of tree amplitudes -- to loop-level integrands, facilitating the use of the tree-level double copy to generate complete gravitational integrands at any loop order. This method circumvents the difficulties in identifying complete higher-loop-order gauge-theory integrands that adhere to the color-kinematics duality. Additionally, our cut-based organization is well-suited for expansion in hard or soft limits, aiding in the exploration of ultraviolet or classical limits of scattering amplitudes.